
where \(\theta ^T \overline{w}_{S_\phi (x)}\)  and \(\theta ^T \underline{w}_{S_\phi }\)  are the linear upper/lower bound of \(f_{\theta }(x^{\prime })\)  inside the set \(S_\phi (x) = \lbrace x^{\prime }\in D_\phi | d(x,x^{\prime }) = 1\rbrace \) . Such linear upper/lower bound of \(f_{\theta }(x^{\prime })\)  can be computed using off-the-shelf bounding techniques like Symbolic Interval Analysis [1]} and \(\alpha \) -Crown [2]}. We denote by \(\overline{\mathcal {L}}(y; \theta )\)  the right hand side of the inequality in (REF ) which depends only on the label \(y\)  and the NN parameters \(\theta \) . Now the fairness property can be incorporated in training by optimizing the following problem over \(\theta \)  (the NN parameters):
\(\min _{\theta } \mathop {\mathbb {E}}_{(x,y) \in (X,y)} \left[(1-\lambda _f)\underbrace{\mathcal {L}(f_\theta (x),y;\theta )}_{\text{natural loss}} +\lambda _f \underbrace{\overline{\mathcal {L}}(y; \theta )}_{\text{local fairness loss}} \right]\text{,}\) 
